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6. Let E be a separable Hilbert space and T a continuous operator on E Let R and L
be the positive hermitian operators which are the square roots (15.1112 of r*T and
TT\ respectively. We write R - abs(D, and call it the absolute value of T. Then

Z, = 'abs(T*).                                           ___     -----

(a) Show that Ker(D = KerGR) and that L(E) = T(E). There exists a unique isometry
Vof R(E) onto T(E) such that T= VR. If we extend Vby continuity to R(E), and then
to an operator Ue *(£) by putting U(x) = 0 on the orthogonal supplement of
then we have also T=UR (polar decomposition of T). Show that

R = CW= U*UR = RU*U,       L = URU",       T= LU*.

fb)   For T to be invertible it is necessary and sufficient that K = abs(T) and
1 = abs(r*) are invertible. (To prove necessity, consider the spectra of R and L. To
prove sufficiency, use the closed graph theorem.)                                     ......

(c)   JVis normal if and only if abs(AO = absGV*), and if this condition is satisfied there
exists a unitary operator W such that N = W- abs(AO.

7    A compact operator T on a separable Hilbert space E is said1 nuclear if denoting
by (A/the full sequence of eigenvalues of abs(T) (Section 11.5, Problem 8), we have

a) Use polar decomposition (Problem 6) to show that the product SA of two
Hilbert-Schmidt operators is nuclear. Conversely, if T is nuclear, then abs r)>"
is a self-adjoint Hilbert-Schmidt operator, and T is the product of two Hilbert-
Schmidt operators. Consequently T* is also nuclear. If A is any continuous operator
on E, then AT and TA are nuclear.                                                                        .

(b) If A and B are two Hilbert-Schmidt operators and if <>„) is a Hilbert basis
of E, then the series£ (A3 • *n I *„) and^ (BA • en \ *„) are absolutely convergent, and
their sums are equal."(Write B • e, =£ (B • e. \ e»)em.) Consequently, for every unitary
operator t/and every nuclear operator T, we have

£ (U~1TU• en | ea) =£ (T-en\ ea).

n                                                             "

Deduce that, for a nuclear operator T, the sum £ (T- <?„ | O >s independent of the
Hilbert basis (*„) chosen. This sum is called the trace of 7" and is denoted by
Tr(T). If A, B are two Hilbert-Schmidt operators, then Tr(^-B) = Tr(BA) = (A \ B*).
(c) If T is nuclear, show that

Tr(abs(r» = supfeK3"' a» 16»>l)

where the supremum is taken over all pairs of Hilbert bases (an), (ba) of E
Se the polar decomposition of E). If we put \\T\\t = Tr(abs(D), then the set ^(E)
of nuclear operators on E is a vector space on which ||r||, is a norm, such that

II T"ll   <I II Til

d)   I?(T.Hs a sequence of nuclear (resp. Hilbert-Schmidt) operators on E which
converges weakly (Section 12.15, Problem 9) to an operator T, and which is such that
the sequence of norms (||r,|U) (resp. (||r,||a)) is bounded, then Tis a nuclear (resp,
Hilbert-Schmidt) operator,
(e)   Show that ^?i(E) is a Banach space with respect to the norm || / ||,.quently u — a + ib is unitary, and a =